Box plots

The ages of 7 students in a drama workshop have been recorded: $12, 14, 15, 18, 20, 22, 25$.

Calculate the minimum, the first quartile ($Q_1$), the median ($Me$), the third quartile ($Q_3$) and the maximum. Also determine the interquartile range ($IQR$).

Given the following list of temperatures in degrees Celsius recorded over 8 days, in no particular order: $15, 12, 19, 14, 18, 10, 21, 16$.

Order the data from smallest to largest and calculate the five-number summary required to draw a box plot.

In a study on the weight of school backpacks, it was found that the first quartile is $Q_1 = 4,5$ kg and the third quartile is $Q_3 = 7,2$ kg.

Calculate the interquartile range ($IQR$). If the minimum weight recorded is $2$ kg, what is the length of the left whisker in the box plot?

State whether the following statements are true or false, justifying your answer in each case:

• The line marking the median is always located at the exact geometric centre of the box.

• The two whiskers of a box plot always have the same length.

• The interquartile range is not affected by extreme values (maximum and minimum).

The salaries (in euros) of 5 employees in a small company are: $1100, 1200, 1300, 1500, 2500$.

Calculate the five-number summary. If the employee earning $2500$ € receives a pay rise of $500$ € (now earning $3000$ €), how does this affect the median and the interquartile range? What advantage does the box plot have over the arithmetic mean in this case?

In a statistical distribution, the first quartile is $Q_1 = 40$ and the third quartile is $Q_3 = 64$.

Calculate the interquartile range ($IQR$). Next, determine the lower and upper limits to identify outliers (using the $1,5 \cdot IQR$ rule). Would the value $105$ be considered an outlier? What about the value $3$?

The heart rate (in beats per minute) of 11 athletes after a race has been recorded: $110, 115, 120, 122, 125, 130, 135, 140, 145, 150, 200$.

Calculate the five-number summary and identify whether the value $200$ is an outlier using the $1,5 \cdot IQR$ criterion. If we remove this outlier from the data set, how do the mean and median change? Which measure of central tendency is more robust against this change?

The following graph shows the box plots of resting heart rates in beats per minute for two groups of athletes, A (distance runners) and B (weightlifters).

Based on observing the graphs, answer the following:

• Which group has a lower median?

• Which group shows greater spread in the central $50\%$ of its data?

• Calculate the interquartile range for both groups and compare them.

A box plot has the following values: Minimum = $10$, $Q_1 = 15$, $Me = 18$, $Q_3 = 35$, Maximum = $40$.

Analyse the skewness of the distribution. Is it skewed to the right or to the left? Justify your answer by comparing the distances between the quartiles and the median.

A company analyses the lifespan (in hours) of two brands of light bulbs, A and B. Both brands have the same median ($Me = 1000$ h) and the same interquartile range ($IQR = 200$ h). However, the overall range for brand A is $400$ h, whereas the overall range for brand B is $1200$ h.

What information does this difference in overall ranges give us regarding the reliability and the presence of outliers in both brands? Which of the two brands would you consider more predictable and why?

The marks for an exam out of 10 points have the following summary: Minimum = $3$, $Q_1 = 5$, $Me = 6$, $Q_3 = 8$, Maximum = $10$.

The teacher decides to transform the marks by multiplying them by $0,8$ and adding $2$ points. Calculate the new five-number summary and the new interquartile range.

In a statistical distribution, the quartiles are given by algebraic expressions in terms of a parameter $x > 0$:

• $Q_1 = x$

• $Me = 2x + 1$

• $Q_3 = x^2 - 2$

Knowing that the interquartile range ($IQR$) is equal to $10$, set up an equation, solve it to find the value of $x$, and calculate the numerical values of $Q_1$, $Me$ and $Q_3$.